Quantum Circuit Tensors and Enumerators with Applications to Quantum Fault Tolerance

TL;DR

This paper introduces a mathematical framework using tensor and circuit enumerators to analyze quantum circuits and error models exactly, enabling efficient error path counting in quantum error correction codes.

Contribution

It extends tensor enumerators to circuit enumerators, providing a novel method for analyzing quantum circuits and error models without Monte Carlo simulations.

Findings

Exact error path counting in a distance five surface code

Development of a Poisson summation analogue for stabilizer codes

Relation of circuit enumerator to process matrix via MacWilliams identity

Abstract

We extend the recently introduced notion of tensor enumerator to the circuit enumerator. We provide a mathematical framework that offers a novel method for analyzing circuits and error models without resorting to Monte Carlo techniques. We introduce an analogue of the Poisson summation formula for stabilizer codes, facilitating a method for the exact computation of the number of error paths within the syndrome extraction circuit of the code that does not require direct enumeration. We demonstrate the efficacy of our approach by explicitly providing the number of error paths in a distance five surface code under various error models, a task previously deemed infeasible via simulation. We also show our circuit enumerator is related to the process matrix of a channel through a type of MacWilliams identity.